I came across this interesting paper "Low-Dimensional Hyperbolic Knowledge Graph Embeddings" - Chami et al. . It is about embedding Graph Structures (more concretely: Knowledge Graphs) into Hyperbolic Space (Poincaré Ball model). In the Background section, the authors show the following figure of a hyperbolic manifold (Figure 2):
The caption of the figure says: "... to the hyperbolic manifold". Hyperbolic space has a constant negative curvature. But for me, the manifold on this figure looks like it has positive curvature. I am confused and don't know if I misunderstood the key idea of Hyperbolic space and negative curvature OR if I interpret the figure wrong OR if it is actually not a hyperbolic manifold.
I would really appreciate your help. Not understanding this figure drives me crazy.
Thanks a lot!
This picture, like many pictures, is not intended to be analytically exact. It is instead intended to convey a general intuition. The general intuition here is applicable to any complete Riemannian manifold whatsoever; a hyperbolic manifold is just a special case of a complete Riemannian manifold, namely it is a complete Riemannian manifold of constant sectional curvature $-1$.
The picture could just as well have been used in any Riemannian geometry book, with an alternate caption: "An illustration of the exponential map $\exp_x(v)$, which maps the tangent space $\mathcal T_x M$ at the point $x$ to the complete Riemannian manifold $M$".